DOCUMENTS ON MODERN PHYSICS Edited by w. ELLIOTT MONTROLL, University of Rochester GEORGE H. VINEYARD, Brookhaven National Laboratory MAURICE LEvY, Universite de Paris A. ABRAGAM, L'Effet Mossbauer H. BACRY, Lectures on Group Theory S. T. BELYAEV, Collective Excitations in Nuclei T. A. BRODY, Symbol-Manipulation Techniques for Physics K. G. BUDDEN, Lectures on Magnetoionic Theory J. W. CHAMBERLAIN, Motion of Charged Particles in the Earth's Magnetic Field S. CHAPMAN, Solar Plasma and Geomagnetism and Aurora H. Cmu, Neutrino Astrophysics A. H. CoITRELL, Theory of Crystal Dislocations J. DANON, Lectures on the Mossbauer Effect BRYCE S. DEWITT, Dynamical Theory of Groups and Fields R. H. DICKE, The Theoretical Significance of Experimental Relativity P. FoNG, Statistical Theory of Nuclear Fission M. GOURDIN, Laws of Symmetry Theorem of T. C. P. D. HESTENES, Space-Time Algebras JoHN G. KIRKwooD, Selected Topics in Statistical Mechanics; Macromolecules; Theory of Liquids; Theory of Solutions; Proteins; Quantum Statistics and Cooperative Phenomena; Shock and Detonation Waves; Dielectrics-Intermolecular Forces-Optical Rotation R. LATTES, Methods of Resolutions of Some Boundary Problems in Mathematical Physics F. E. Low, Symmetries and Elementary Particles P. H. E. MEDER, Quantum Statistical Mechanics M. MosJDNsKY, Group Theory and the Many-body Problem M. NIKOLIC, Kinematics and Multiparticle Systems; Analysis of Scattering and Decay A. B. Pn>PARD, The Dynamics of Conduction Electrons L. SCHWARTZ, Applications of Distributions to the Theory of Elementary Particles in Quantum Mechanics J. ScHWINGER, Particles and Sources J. SCHWINGER AND D. SAXON, Discontinuities in Wave Guides M. TINKHAM, Superconductivity Additional Volumes in preparation Particles and Sources JULIAN SCHWINGER Harvard University Notes by Tung-mow Yan GORDON AND BREACH, SCIENCE PUBLISHERS New York London Paris. Copyright© 1969 by Gordon and Breach, Science Publishers, Inc. 150 Fifth Avenue, New York, N. Y. 10011 Library of Congress Catalog Card Number: 69-19390 Editorial office for Great Britain: Gordon and Breach, Science Publishers Ltd. 8 Bloomsbury Way London W. C. 1 Editorial office for France: Gordon & Breach 7-9, rue Emile Dubois Paris 14e Distributed in Canada by: The Ryerson Press 299 Queen Street West Toronto 2B, Ontario All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any in formation storage and retrieval system, without permission in writing from the Publishers. (0206) Printed in Switzerland by City-Druck AG, Zurich Editors' Preface Seventy years ago when the fraternity of physicists was smaller than the audience at a weekly physics colloquium in a major university, a J. Willard Gibbs could, after ten years of thought, summarize his ideas on a subject in a few monumental papers or in a classic treatise. His competition did not intimidate him into a muddled correspondence with his favorite editor nor did it occur to his colleagues that their own progress was retarded by his leisurely publication schedule. Today the dramatic phase of a new branch of physics spans less than a decade and subsides before the definitive treatise is published. Moreover, modern physics is an extremely interconnected discipline and the busy practitioner of one of its branches must be kept aware of breakthroughs in other areas. An expository literature which is clear and timely is needed to relieve him of the burden of wading through tentative and hastily written papers scattered in many journals. To this end we have undertaken the editing of a new series, entitled Docu ments on Modern Physics, which will make available selected reviews, lecture notes, conference proceedings, and important collections of papers in branches of physics of special current interest. Complete coverage of a field will not be a primary aim. Rather, we will emphasize readability, speed of publication, and importance to students and research workers. The books will appear in low-cost paper-covered editions, as well as in cloth covers. The scope will be broad, the style informal. vi EDITORS' PREFACE From time to time, older branches of physics come alive again, and forgotten writings acquire relevance to recent developments. We expect to make a number of such works available by including them in this series along with new works. ELLIOTT W. MONTROLL GEORGE H. VINEYARD MAURICE LEVY Contents 1. NONINTERACTING PARTICLES 1 1.1 Introduction 1.2 Sources 3 1.3 Spinless Particles 4 1.4 An Application. Stimulated Emission 10 1.5 Spin-1 Particles. The Photon 12 1.6 Massless Spin-2 Particles. The Graviton 16 1.7 Spin-(1/2) Particles 18 1.8 Multispinor Formulation of Arbitrary Spin 25 General Connection of Spin and Statistics 2. ELECTRODYNAMICS 29 2.1 Combined System of Noninteracting Electrons and Photons. 29 Fields 2.2 Primitive Interaction and Interaction Skeleton. Extended Sources 32 2.3 Calculation of Matrix Elements 37 2.4 Two-Particle Exchange 39 2.5 Modified Photon Propagation Function 42 2.6 Electromagnetic Form Factors 46 Vlll CONTENTS 3. STRONG AND WEAK INTERACTIONS 53 3.1 Low-Energy re+ N System 53 3.2 Partial Symmetry. Chiral Invariance 56 3.3 Non-Abelian Vector Gauge Particles, p and A 1• 60 3.4 Widths of p and A 68 1 3.5 Low-Energy rcrc Interactions 72 3.6 Spectrum of the Decay Process 17*(960)-->17 + 2rc 76 3.7 Pion Electromagnetic Mass 80 3.8 U x U Partial Symmetry 86 4 4 3.9 Concluding Remarks 91 PARTICLES AND SOURCES 1. NONINTERACTING PARTICLES 1.1 Introduction Research in particle physics presently falls in two main classi fications: Quantum Field Theory and S-Matrix Theory. While quantum field theory, like field theorists, comes in all sizes and shapes we regard its basic characteristics to be as follows. It is a space-time formulated operator theory. The fundamental dynamical variables, the fields, describe certain localized excitations, which in particle language, correspond to all possible combinations of particles with the prescribed quantum numbers. When the physical couplings are weak, as in electrodynamics, the relation between field and particle may appear to be close. For strongly interacting systems it is certainly very remote. Current algebra was originally motivated by a reaction against field theory, in which currents, supposedly more physical, were introduced as basic operators. But the latest development in this area has been a return to the phenomenological field as primary. The practical difficulty in connecting field-theory hypotheses with raw experimental data led to the phenomenological emphasis of S-matrix theory. Here the particle is primary. Attention is concentrated on the momentum space description of various collisions. Dynamical principles are sought in the analytic extension of the momenta to complex values. In these lectures we shall describe a totally new approach to particle physics. It is intermediate in concept between the two previous formulations. It 1